TL;DR Yes it is possible. There are grafting techniques which can be reliably used to save the tree. usually in the case of rodent damage. Source, Am nursery owner, work with trees/grafting regularly.
Edit: Both the xylem and phloem would be reconnected with a bridge graft as long as you line the scion up properly.
As a Nursery owner Ill throw my two cents in.
Yes it is possible, but unlikely if the two parts were simply balanced together. However there are grafting techniques which can reliably save the tree.
It is highly dependent on tree species, age, health, local weather, time of year, and a huge number of other factors. You would need the tree to be cut so thinly that there is zero diameter change between the two halves of the tree. This is nearly impossibly and is why wedge or vernier grafting exist.
You actually only need some of the vascular tissue (cambium, phloem, xylem) to be lined up for success. Obviously more is better but close to half is good enough for survival. There would be damage but the top would live.
That being said there is a technique which would greatly improve the chances of survival. You could bridge the gap. A bridge graft is where you take stems from younger trees of the same species and use them to connect the two separated pieces.
This can even be done in a way where the old wood from the original tree is removed so you have a large void instead of dead wood there. This technique is rarely practiced but is used to save heritage trees which have been damaged by rodents or mechanical damage usually from people mowing the lawn.
The phloem and cambium tissues could potentially form callous tissue and rebond but the xylem (being already dead) would not resulting in an extreme interruption in water flow. More than likely the tree would die.
Edit: cause auto correct
Edit 2: look, I'm not saying grafting is impossible. We wouldn't have desirable apple cultivars and greenhouse tomatoes if it wasn't. All of those are performed on juvenile plants with care and precision. I am saying that if you cut a large adult tree through and through the odds of it healing and carrying on are slim.
It depends greatly on the species. I've used tree ringing as a forest management technique for years. A cedar is always killed by ringing. A black locust will often jump a shoot across the gap, and I will return the next year to find a 2" thick limb growing across the gap with the upper part of the tree thriving. Many species will sprout from a stump and can be coppiced,repeatedly harvesting the regrowth, but if any species could survive what you described, I would bet on black locust.
It might be possible if you could lift the tree up and dress the cut area with growth hormone powder. This is a common practice done with other perennials over 40+ years and would probably have some success in the scenario that you're describing.
Weeping cherry trees are often grafted to a normal cherry tree. They have been available in nurseries for decades.
Had this happen. Hired a guy to paint the soffits on my house. He couldn't figure out how to get his ladder in a good place, so he cut down a decorative tree we had tended for years.
Came home, saw most of my tree lying in the yard, fired the guy on the spot. He had my number and was to call me for anything like this. So, done.
Grafted the trunks back together by carving down a bit to expose fresh vascular tissue, notched top and bottom so they would hold tightly (think tongue in groove notches), wrapped the graft in plastic and kept it moist (not wet) while it healed.
About 50% of the grafts took. And the tree has now fully regrown but it took a couple of years to get back to its original glory.